Compression Members MCQ with answer and solution

Concept

Slenderness ratio limit for Euler formula

Formula

λ = √(π² × E / σ)

Step-by-step solution

1. Euler's formula is applicable only when buckling stress is less than or equal to the crushing stress (3300 kg/cm²). 2. Euler buckling stress formula is σ = (π² × E) / (λ²), taking E = 2.05 x 10⁶ kg/cm² for mild steel. 3. Setting (π² × 2.05 x 10⁶) / (λ²) ≤ 3300 yields λ² ≥ 6130, giving slenderness ratio λ ≥ 80. Answer: D

Explanation

Euler's formula applies strictly to elastic buckling and remains valid only when the critical buckling stress does not exceed the material's yield or crushing strength. For a mild steel column with a crushing stress of 3300 kg/cm², setting the Euler stress equal to this limit yields a minimum slenderness ratio of 80. Below a slenderness ratio of 80, the column fails by direct crushing rather than elastic buckling.

Why the other options are wrong

A) At a slenderness ratio of 40, direct crushing dominates and Euler’s formula is not applicable. B) A slenderness ratio of 50 is below the critical slenderness ratio for Euler’s formula applicability in mild steel. C) A slenderness ratio of 60 falls in the short/intermediate column range where Rankine’s formula is preferred over Euler’s.

Shortcut

Mild steel Euler formula applies when slenderness ratio λ ≥ 80.

Common mistake

Applying Euler's formula without verifying if the slenderness ratio exceeds the critical threshold for mild steel.